Additional file 3: Calculating Intra Class Correlations (ICCs)
In linear regression models the ICC for a higher level (in our study: insurance physician and
office level) can be calculated by using the difference between the error variance of the same
models with and without random coefficients. Unfortunately, no total variance is given (in
MLwin) for logistic and Poisson regression models.
There is no commonly accepted method for calculating ICCs in logistic and Poisson
regression models [29]. One option is to use the estimated variance coefficient of the random
intercept for a level, in which case the formula is: the between-error variance (σ2) divided by
the sum of the between-error variance (σ2) and the intra-cluster variance, i.e. a third of the
squared pi (π2/3), i.e. σ2/[σ2 + (π2/3)] [29, 40]. The between-error variance for a level is
estimated with the variance coefficient of the random intercept for that level. For example, the
between-error variance for insurance physician level is estimated in a model with a random
intercept for IP level alone and with only fixed coefficients for the client background
variables; hence, no IP variables. The between-error variance for office level is then estimated
with the same model, but with a random intercept both for IP and office level.
As this calculation method is difficult to interpret when determining an ICC for a
logistic and Poisson regression model [29], we chose a method that we considered more
appropriate for the goal of this study, one that would take explicit account of the associations
between the client variables and IP and office level. We therefore calculated the individual
outcome measures of an estimated model using the prediction option in MLwin. First, we
calculated the predicted outcome for the full model with random coefficients (intercept and/or
slopes) for both office level and IP level, then the predicted outcome for an intermediate
model with only random coefficients for IP level, and finally, for the model with only fixed
coefficients for intercept and client background variables. All these predictions were made
under ceteris paribus conditions, starting from the full model.
The predicted outcome measures took the form of logit(ye) for the logistic regression
models and the form of ln(ye) for the Poisson regression models. We then transformed the
predicted outcome measure ye =1/(1+e-z) for the logistic model with z=logit(ye), and ye=ez for
the Poisson model with z= ln(ye) [28]. By calculating the squared correlation between the
observed outcome measure y and the predicted outcome measure ye, we obtained the
explained variance (R2) of each model. Hence, in the case of two models, e.g. model A with
random coefficients for insurance physicians and model B with fixed coefficients, the ‘error
variance’ is σ2ea=(1-Ra2) and σ2eb=(1-Rb2), with σ2eb ≥ σ2ea . We calculated the ICC for the
level in question as (σ2eb- σ2ea)/ σ2eb. Here, the ICC for office level was calculated as the
difference in the error variances between the full model, including random coefficients for
office level and IP level, and the model with only random coefficients for IP level. The ICC
for IP level was calculated as the difference in the error variances between the model with
random coefficients for IP level and the model with only fixed coefficients for the client
background variables. This procedure enabled us to take account of the possible influence of
IP variables on the ICC at IP level.